In this paper, we study a non-Lipschitz and box-constrained model for both piecewise constant and natural image restoration with Gaussian noise removal. It consists of non-Lipschitz isotropic first-order \(\ell _{p}\) (\(0<p<1\)) and second-order \(\ell _{1}\) as regularization terms to keep edges and overcome staircase effects in smooth regions simultaneously. The model thus combines the advantages of non-Lipschitz and high order regularization, as well as box constraints. Although this model is quite complicated to analyze, we establish a motivating theorem, which induces an iterative support shrinking algorithm with proximal linearization. This algorithm can be easily implemented and is globally convergent. In the convergence analysis, a key step is to prove a lower bound theory for the nonzero entries of the gradient of the iterative sequence. This theory also provides a theoretical guarantee for the edge preserving property of the algorithm. Extensive numerical experiments and comparisons indicate that our method is very effective for both piecewise constant and natural image restoration.

在本文中，我们研究了具有高斯噪声消除功能的分段恒定和自然图像恢复的非Lipschitz模型和盒约束模型。它由非Lipschitz各向同性一阶\（\ ell _ {p} \）（\（0 <p <1 \））和二阶\（\ ell _ {1} \）组成作为正则化术语，可以同时在平滑区域中保持边缘并克服阶梯效应。因此，该模型结合了非Lipschitz和高阶正则化的优点以及框约束。尽管此模型的分析非常复杂，但我们建立了一个激励定理，该定理引发了一种具有近端线性化的迭代支持收缩算法。该算法可以轻松实现，并且可以全局收敛。在收敛性分析中，关键步骤是证明迭代序列梯度的非零项的下界理论。该理论也为算法的边缘保持性提供了理论保证。大量的数值实验和比较表明，我们的方法对于分段恒定和自然图像恢复都是非常有效的。